Boltzmann-Gibbs entropy is sufficient but not necessary for the likelihood factorization required by Einstein

Abstract

In 1910 Einstein published a work on a crucial aspect of his understanding of the Boltzmann entropy. He essentially argued that the likelihood function of any system composed by two probabilistically independent subsystems ought to be factorizable into the likelihood functions of each of the subsystems. Consistently he was satisfied by the fact that the Boltzmann (additive) entropy fulfills this epistemologically fundamental requirement. We show here that entropies (e.g., the q-entropy on which nonextensive statistical mechanics is based) which generalize the BG one through violation of its well-known additivity can also fulfill the same requirement. This important fact sheds light on the very foundations of the connection between the micro- and macro-scopic worlds, and consistently supports that the classical thermodynamical Legendre structure is more powerful than the role to it reserved by the Boltzmann-Gibbs statistical mechanics.